PHYSICAL REVIEW E 68, 016409 共2003兲

Angular momenta creation in relativistic electron-positron plasma
T. Tatsuno
Graduate School of Frontier Sciences, The University of Tokyo, Tokyo 113-0033, Japan

V. I. Berezhiani
Institute of Physics, The Georgian Academy of Sciences, Tbilisi 380077, Georgia

M. Pekker and S. M. Mahajan
Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712, USA
共Received 22 July 2002; published 23 July 2003兲
Creation of angular momentum in a relativistic electron-positron plasma is explored. It is shown that a chainof angular momentum carrying vortices is a robust asymptotic state sustained by the generalized nonlinear
Schrödinger equation characteristic to the system. The results may suggest a possible electromagnetic origin of
angular momenta when it is applied to the MeV epoch of the early Universe.
DOI: 10.1103/PhysRevE.68.016409

PACS number共s兲: 52.40.Db, 52.27.Ny

I. INTRODUCTION

seem to be essential constituents of the Universe, and 共2兲
under certain conditions, even an ultrarelativistic electronproton plasma can behave akin to an e-p plasma 关9兴.
Recently, we found dark soliton as well as vortex soliton
solutions in e-p plasmas 关10兴. In Ref. 关11兴, it is also shown
analytically that the dark soliton is the natural nonlinear coherent structure in unmagnetized cold e-p plasmas. However, it is conceivable that soliton solutions obtained in a
one-dimensional formulation will turn out to be unstable inhigher dimensions, which may lead to the creation of vortex
solitons. Since dark and vortex solitons are asymptoticallynonvanishing, they have received much less attention thantheir localized cousins due to the generally accepted requirement that the fields be localized in a physical system. However, in recent experiments studying laser field dynamics indifferent kinds of optical media, it was demonstrated thatdark and vortex solitons can be readily created as superimpositions upon a localized field background 关12兴. Vortex soliton solutions have been also found in imperfect Bose gas in
the superfluids 关13兴 and have been extensively investigatedand discussed 关14兴. In this paper, we systematically investigate the instability of dark solitons and show that it can leadto the formation of vortex solitons. We speculate about aninteresting application of the vortex soliton in the early
Universe.
In a recent paper 关10兴, we had developed an argument forthe creation of domains of nonzero angular momentum in the
MeV era of the early Universe when it was supposed to be
dominated by a plasma of e-p pairs. We first showed that insuch a plasma the dynamics of a pulse of electromagnetic
radiation, with a frequency much larger than the plasma frequency, is controlled by a generalized nonlinear Schrödinger
equation 共GNSE兲 with a defocusing nonlinearity. Then, borrowing a result of nonlinear optics 关12兴, where the standard
nonlinear Schrödinger equation 共NSE兲 with a cubic nonlinearity has been investigated in great depth and detail, we
conjectured that even the GNSE, whose nonlinearity is similar in nature to that of NSE, will allow vortex soliton solutions with an angular momentum that is conserved during
propagation. The latter system also allows dark soliton solu-

The problem of electromagnetic 共EM兲 wave propagation
and related phenomena in relativistic plasmas has attracted
considerable attention in the recent past. From the nonthermal emission of the high-energy radiation coming from avariety of compact astrophysical objects it has become possible to deduce the presence of a population of relativistic
electrons in the plasma created in the dense radiation fieldsof those sources 关1兴. The principal components of these plasmas could be either relativistic electrons and nonrelativisticions 共protons兲 or relativistic electron-positron (e-p) pairs.
Relativistic e-p dominated plasmas may be created in avariety of astrophysical situations. The e-p plasmas arelikely to be found in pulsar magnetospheres 关2兴, in the bipolar outflows 共jets兲 in active galactic nuclei 关3兴, and at thecenter of our own galaxy 关4兴. The presence of e-p plasma isalso argued in the MeV epoch of the early Universe. In thestandard cosmological model, temperatures in the MeV
range (T⬃1010 K⫺1 MeV) prevail up to times t⫽1 sec after the Big Bang 关5兴. In this epoch, the main constituent of
the Universe is the relativistic e-p plasma in equilibriumwith photons, neutrinos, and antineutrinos.
Contemporary progress in the development of superstrong laser pulses with intensities I⬃1021⫺23 W/cm2 has
also made it possible to create relativistic plasmas in thelaboratory by a host of experimental techniques 关6兴. At thefocus of an ultrastrong laser pulse, the electrons can acquire
velocities close to the speed of light, opening the possibilityof simulating in the laboratory the conditions and phenomena that, generally, belong to the astrophysical realm 关7兴.
Elucidation of the electromagnetic wave dynamics in a
relativistic plasma will, perhaps, be an essential tool for understanding the radiation properties of astrophysical objects
as well as of the media exposed to the field of superstrong
laser radiation. Although the study of wave propagation in
relativistic plasmas has been in vogue for some time, it is
only in the recent years that the nonlinear dynamics of EM
radiation in e-p dominated plasmas 关8兴 has come into focus.
The enhanced interest stems from two facts: 共1兲 e-p plasmas
1063-651X/2003/68共1兲/016409共8兲/$20.00

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PHYSICAL REVIEW E 68, 016409 共2003兲

TATSUNO et al.

tions in one dimension which are known to be unstable totwo-dimensional 共2D兲 perturbations and eventually evolve to
2D vortex chains.
In this paper we demonstrate that the arguments given in
Ref. 关10兴 are actually borne out by the direct solutions of thederived GNSE. We begin by analytically showing that the
1D dark solitons of the GNSE are, indeed, unstable to 2D
perturbations. In the process, we derive the instability criterion which agrees with the numerical solution of the linearized system.
The main part of the demonstration, however, comes froma numerical simulation of our GNSE. Starting from a broad
variety of initial conditions, we find the eventual emergenceof angular momentum carrying vortex solitons. That is, thestipulated solitonic structures are readily accessible within
the framework of this general equation with a nonlinearity
more complicated than that of NSE. Thus we can state withmuch greater confidence that electromagnetism, operative in
the MeV era, could easily be the primordial source of angular momentum associated with various structures of the observable universe.
We again stress that the present results are quite general in
the nonlinear dynamics in an electron-positron plasma.

and n⬃1030 –1033 cm⫺3 ), the radiation pressure ␴ T 4 issmall compared to the kinetic pressure nT, and was neglected in Eq. 共2兲; inclusion of the photon gas in the system
only leads to a small modification of the effective ‘‘mass’’ ofparticles.
The low-frequency motion of the plasma is driven by the
ponderomotive pressure 关 ⬃(pe,p ) 2 兴 of the high-frequency
EM field, and is independent of the sign of the particles’
charge. It is perfectly natural to assume that the electron andthe positron fluids have equal temperatures (T 0e,p ⫽T 0 ) in
equilibrium so that their effective masses (G e,p ⫽G) will
also be equal. The radiation pressure will impart equal lowfrequency momenta to both fluids allowing the possibility ofoverall density changes without producing charge separation.
The charge neutrality conditions n e ⫽n p ⫽n, ␾ ⫽0 havebeen assumed by neglecting the small inequality of thecharge due to baryon asymmetry. It is also evident that thesymmetry between the two fluids keeps their temperatures
always equal (T e,p ⫽T) if they were equal initially. In deriving Eqs. 共1兲 and 共2兲, we have also assumed that the plasma istransparent 共i.e., ␻ Ⰷ ␻ e ), and that the longitudinal extent of
the pulse is much shorter than its transverse dimensions(L 储 ⰆL⬜ ).
Defining the normalized variables

II. SINGLE-VORTEX SOLITON

In the envelope approximation, a finite amplitude, circularly polarized EM pulse propagating in a relativistic e-p
plasma obeys 共for details, see Ref. 关10兴兲
2i ␻⳵ ␶ A⬜ ⫹c 2 ⵜ⬜2 A⬜ ⫹

c2

␥ 2g

⳵ 2␰ A⬜ ⫹2 ␻ 2e
G0

冉

冊

G0 n
1⫺
A ⫽0,
␥G n0 ⬜
共1兲

where A⬜ is the slowly varying amplitude of the perpendicular 共to the propagation direction兲 vector potential, ⵜ⬜2 ⫽ ⳵ 2x
⫹ ⳵ 2y , and ␰ ⫽z⫺ v g t is the ‘‘comoving’’ 共with group velocity v g ) coordinate with ␶ ⫽t. Here ␻ e ⫽(4 ␲ n 0e e 2 /m e ) 1/2 isthe electron Langmuir frequency, n 0 ⫽n 0e ⫽n 0p is the number density of the unperturbed background 共subscript 0 denotes the value at infinity兲, ␥ is the Lorentz factor, and G⫽K 3 (m e c 2 /T)/K 2 (m e c 2 /T) with K n , denoting the nth-ordermodified Bessel functions of the second kind, is the enthalpy
density.
In this approximation, the continuity equation, determining the number density n, becomes
1 nv g ␥ g /c
⫽ 2 2
,
2
␥ G n 0 关 ␥ g G 0 ⫺G ⫺e 2 兩 A⬜ 兩 2 / 共 m e c 2 兲 2 兴 1/2

␥ K 2 共 m e c 2 /T 兲

exp共 ⫺m e c G/T 兲 ⫽const.
2i ⳵ t A⫹ 12 ⵜ⬜2 A⫹ f 共 兩 A 兩 2 兲 A⫽0,

共5兲with the nonlinearityf 共 兩 A 兩 2 兲 ⫽⫺2

冉

1

冑1⫺ 兩 A 兩 2

冊

⫺1 .

共6兲

Note here that 兩 A 兩 ⬍1 is necessary for the regularity of n 关see
Eq. 共2兲兴. According to Eq. 共14兲 of our previous paper 关10兴,
momentum p becomes complex when 兩 A 兩 ⬎1, implying themultistream motion of the fluid.
Equation 共5兲 admits a symmetric two-dimensional solitary
wave solution. For stationary solitons, the ansatz
A⫽A 共 r 兲 exp共 im ␪ ⫺i␭t 兲

共7兲

reduces Eq. 共5兲 tod2

共3兲Here, just like Ref. 关15兴, we have assumed that the temperature variation in the system is negligibly small. Furthermore,in the density and temperature range of interest (T⯝m e c 2

共4兲

and noting that even with the assumption ⳵ ␰ Ⰷⵜ⬜ , the diffractive term can be of the same order or even greater thanthe dispersive one for a highly transparent plasma ( ␥ g Ⰷ1),
Eq. 共1兲 converts to the following GNSE ( ␥ g G 0 ⰇG):

共2兲and the system is closed by the adiabatic equation of state
nm e c 2 /n 0 T

␻ 2e
␻e
1
e
t→t,
r→r,
A⬜ →A⬜ ,2
2␻G0␥
c 冑2G 0m ec gG 0

1 dA m 2⫹ 2 A,r dr
r共8兲

V 共 A 兲 ⫽ 共 ␭⫹2 兲 A 2 ⫹4 冑1⫺A 2 ⫺4,

共9兲

dr

A⫹V ⬘ 共 A 兲 ⫽⫺2

with potential V(A) defined as

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PHYSICAL REVIEW E 68, 016409 共2003兲

ANGULAR MOMENTA CREATION IN RELATIVISTIC . . .

with the prime denoting the derivative with respect to its
argument, and ␭ representing the nonlinear frequency shift.
A numerical analysis of the solution of this eigenvalue
problem is given in Ref. 关10兴. Because of the absence of ␰
derivatives in Eq. 共5兲, any pulselike localized structure isallowed in the propagation direction. In the direction perpendicular to the propagation, on the other hand, the amplitude
was shown to approach a constant value. Since there does
exist a small baryon asymmetry in the early Universe thatacts on a longer characteristic length 关15,16兴, the extent of
the constant amplitude region may be considered to be finite.
Due to the single valuedness of the vector potential, mmust be an integer and A must vanish at the origin for nonzero values of m. The nonzero m solutions are particularly
important because they carry the orbital angular momentumM,

共 M兲 z ⫽i2冕dr⬜ 关 r⬜ ⫻ 共 A * “⬜ A⫺c.c.兲兴 z .

Since we are examining the linear stability of the solution
with the least number of nodes, g is assumed to be an oddfunction with a single node at the origin.
Let us perturb the one-dimensional solution byA⫽ 共 g⫹u⫹i v 兲 e ⫺i␭t ,where the small perturbations u and v are real functions of x
and y.
Assuming sinusoidal behavior, u⬀cos(ky⫺⍀t) and v⬀sin(ky⫺⍀t), the linearized GNSE could be reduced to thefollowing eigenvalue problem:
k2
k4
共 L0 ⫹L1 兲v ⫹ v ,
2
4

共13兲

1 ⳵2
⫹ f 共 g2兲,
2 ⳵x2

共14兲

1 ⳵2
⫹ f 共 g 2 兲 ⫹2g 2 f ⬘ 共 g 2 兲 .
2 ⳵x2

共15兲

⍀ 2 v ⫽L1 L0 v ⫺
where

共10兲

It is straightforward to show that system 共5兲 conserves angular momentum, and expression 共10兲 is just the paraxial approximation for the orbital angular momentum, ME ⫽ 兰 dr关 r⫻(E⫻B) 兴 , of the EM field 关17兴. The angular momentum
carried by the vortex is M z ⫽mN, where the photon numberN⫽ 兰 dr⬜ 兩 A 兩 2 is another conserved quantity; m is also known
as the ‘‘topological charge.’’ Strictly speaking, one must redefine the integrals of motion for nonvanishing boundary
conditions 关18兴, but such a renormalization is not important
here because of the fact that an infinite-extent solution is justa formal approximation. The presence of a small fraction ofions makes the physical solution decay at infinity 关15,16兴.

共12兲

L0 ⫽␭⫹

L1 ⫽␭⫹

Remembering that L0 g⫽0 and L1 g x ⫽0, we can construct from Eq. 共13兲 the relation⍀ 2具 g x兩 v 典 ⫽k4
k2
具 g x 兩 v 典 ⫹ 具 g x 兩 L1 L0 v 典 ⫺ 具 g x 兩 2L1 v 典
4
2
⫺

k2
具 g 兩 ⫺2g 2 f ⬘ v 典
2 x

共16兲

by multiplying g x on both sides and integrating with respect
to x. Self-adjointness of L1 reduces Eq. 共16兲 toIII. DYNAMICS OF ANGULAR MOMENTA CREATION

It is already suggested in Ref. 关10兴 that dark stripe solitons are unstable and break into vortex filaments. When theamplitude is small ( 兩 A 兩 Ⰶ1), the nonlinearity reduces tosimple Kerr-type, and the vortex dynamics of NSE with sucha nonlinearity has been studied in a variety of numericalcalculations 关12,18 –20兴. Our full nonlinearity is somewhat
different and has to be independently investigated. For thissection, our goal is to show that the one-dimensional dark
stripe solution is unstable against transverse perturbations in
two dimensions. Such an instability causes the breakup of
the stripe, leading to a chain of vortex solitons with alternating polarity.
A. Transverse instability

The one-dimensional stationary solution for the GNSE 共5兲
has the form g(x)e ⫺i␭t (␭⬎0), where the real function g(x)satisfies
␭g⫹

1 d 2g⫹ f 共 g 2 兲 g⫽0.
2 dx 2⍀ 2⫽

k 4 k 2 具 g x 兩 ⫺2g 2 f ⬘ v 典⫺,
4
2
具 g x兩 v 典

共17兲

provided 具 g x 兩 v 典 exists. A necessary condition for an exponential instability, then, is

具 g x 兩 ⫺2g 2 f ⬘ v 典⬎0.
具 g x兩 v 典

共18兲

If it is true, then, for sufficiently small 兩 k 兩 , we would have an
instability, i.e., Eq. 共18兲 is also a sufficient condition. Moreover, it is explicitly shown that ␻ →0 in the limit k→0 unless 具 g x 兩 v 典 diverges.
We now make an approximate estimate of the ratio on theleft hand side of Eq. 共18兲. Let us assume that operators L0
and L1 are of order unity and ⍀ 2 and k 2 are, in some sense,
small. In this limit, we could estimate v by solving
L1 L0 v ⫽0,

共19兲

which has the solution v ⫽g⫹q, where
共11兲
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L0 q⫽g x .

共20兲PHYSICAL REVIEW E 68, 016409 共2003兲

TATSUNO et al.

Notice that, for v ⫽g, the integrals occurring in Eq. 共17兲 arezero and this part will not contribute to the integral; only q
will. From Eq. 共20兲, we can deduce the following. The first
consequence具 g x 兩 g x 典 ⫽ 具 g x 兩 L0 q 典 ⫽ 具 g x 兩 共 L0 ⫺L1 兲 q 典 ⫽ 具 g x 兩 ⫺2g 2 f ⬘ q 典⫽ 具 g x 兩 ⫺2g 2 f ⬘ v 典

共21兲

converts Eq. 共17兲 into
k 4 k 2 具 g x兩 g x典
.
⍀ ⫽ ⫺
4
2 具 g x兩 v 典
2

To derive the second, we notice that具 g x 兩 v 典 ⫽ 具 g x 兩 q 典 ⫽ 具 g x 兩 L ⫺1
0 g x典 .

共23兲

In principle, one can evaluate Eq. 共23兲 in a rather straightforward way. But a very approximate estimate can be madeby simply dropping ⳵ 2x in L0 so that冓冏 冔

具 g x兩 v 典 ⯝ g x

gx
␭⫹ f

k4 k2
k4
具 g x兩 g x典
⬅ ⫺ ␣ k 2.
⫺
4
2 具 g x 兩 g x / 共 ␭⫹ f 兲 典 4

共25兲

In Eq. 共25兲 we have overestimated the denominator. The eigenvalue is冋⍀⫽i ␣ k 1⫺

册

k 2 1/2,
4␣⫽0). It is clear from Eq. 共28兲 that the complex conjugate of
the eigenfunction is also an eigenfunction; namely, if w e isan eigenfunction of Eq. 共28兲 corresponding to eigenvalue⍀ e , then w e* is also an eigenfunction of Eq. 共28兲 corresponding to eigenvalue ⍀ *e , where the asterisk denotes the complex conjugate.
For x→⫹⬁, g approaches a constant value g 0 , which canbe expressed as

共24兲

and
⍀ 2⫽

FIG. 1. Dispersion relation for ␭⫽0.2, 0.5, and 1.

共22兲

共26兲

implying a window 0⬍k⬍2 ␣ 1/2 in k, where instability ispossible. From the property that the one-dimensional solution is stable to one-dimensional perturbation, the growthrate ⍀ approaches zero as the wave number k tends to zero.
In order to accomplish the numerical analysis, we introduce a complex-valued function w⫽u⫹i v in Eq. 共12兲. Then,the linearized equation looks

g 0⫽

冑 冉 冊
1⫺

2 2
,
␭⫹2

共29兲

where the inhomogeneity due to the potential vanishes. In
this region, x also becomes an ignorable direction, and we
can assume w⬀e ⫺ ␬ x for the point spectra ( ␬ ⬎0). Plugging
it into our eigenvalue problem, we obtain the following ‘‘dispersion relation,’’ which is applicable for large x:⫺⍀ 2i ⫽ 关 ⍀ r ⫹ 21 共 ␬ 2 ⫺k 2 兲兴关 ⍀ r ⫹ 21 共 ␬ 2 ⫺k 2 兲 ⫹2g 20 f ⬘ 共 g 20 兲兴 ,共30兲
where the subscripts r and i denote the real and imaginary part, respectively, and we have used the relationf (g 20 )⫽⫺␭. By solving it for ␬ , we obtain␬ ⫽ 兵 k 2 ⫺2⍀ r ⫺2g 20 f ⬘ 共 g 20 兲 ⫾2 冑g 40 关 f ⬘ 共 g 20 兲兴 2 ⫺⍀ 2i 其 1/2,
共31兲
where f ⬘ (g 2 ) is explicitly shown asf ⬘ 共 u 兲 ⫽⫺ 共 1⫺u 兲 ⫺3/2.

⫺i ⳵ t w⫽␭w⫹

12

ⵜ⬜2 w⫹ f 共 g 2 兲 w⫹2g 2 f ⬘ 共 g 2 兲 Re共 w 兲 ,共27兲

The asymptotic argument of w is also determined bywhere the prime denotes derivative with respect to its argument. Putting ⳵ t ⫽⫺i⍀ and ⳵ y ⫽ik yields a linear eigenvalue
equation for w:

冉冊

1 d 2w⫺k 2 w ⫹ f 共 g 2 兲 w⫹2g 2 f ⬘ 共 g 2 兲 Re共 w 兲 .
⫺⍀w⫽␭w⫹
2 dx 2共28兲
We have numerically solved Eq. 共28兲 by the shooting methodwith boundary conditions w→0 (x→⬁) and dw/dx⫽0 (x共32兲

arg共 w 兲 ⫽arctan冉2⍀ ik 2 ⫹2⍀ r ⫺ ␬ 2

冊

.

共33兲

With this information, we have solved Eq. 共28兲 numerically. First, we fix an arbitrary value on the real part of w atx⫽10. For an expected eigenvalue, the imaginary part of w
is calculated from Eq. 共33兲. Then, the derivative dw/dx is
estimated by using Eq. 共31兲, and Eq. 共28兲 integrated from x⫽10 to x⫽0 by the fourth-order Runge-Kutta formula. The
data of g(x) is drawn from the numerical solution of Eq.

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ANGULAR MOMENTA CREATION IN RELATIVISTIC . . .

FIG. 3. Zero lines of real and imaginary parts of the field attime t⫽2.

becomes wider and wider as k→0, which agrees qualitatively with the analytical estimate. Since we are performing
the numerical shooting in a finite domain, it becomes difficult to estimate the correct eigenvalue in this regime.
B. Nonlinear evolution

FIG. 2. Eigenfunctions for ␭⫽0.5 corresponding to k⫽0.1, 0.5,
and 0.7.

共11兲. At x⫽0, the boundary condition dw/dx⫽0 is checked.
If this boundary condition is not satisfied, we guess the next
eigenvalue by Newton’s method and carry out the shooting
again.
The dispersion relation is shown in Fig. 1. We can see agood qualitative agreement with the analytic evaluation withthe fact that the growth rate begins from zero as k→0, experiences a maximum value with respect to k, and the modeis finally stabilized for a sufficiently large k. The eigenfunctions corresponding to k⫽0.1, 0.5, and 0.7 for the parameter␭⫽0.5 are illustrated in Fig. 2. From Fig. 1, the mode withk⫽0.5 gives the maximum growth rate Im(⍀)⯝0.137. As isseen from Fig. 2, the imaginary part of the eigenfunction

We will now present the numerical simulation of the dynamics of angular momentum creation from one-dimensional
dark stripe solitons by solving the full nonlinear systems 共5兲
and 共6兲. The calculation was carried out on a 200⫻200 spatial mesh placed in a calculation box of size L⫽28. The
boundary conditions ⳵ x A⫽0 ( ⳵ y A⫽0) are imposed at theedges x⫽⫾L (y⫽⫾L). Note that these boundary conditions numerically respect the constants of motion in the domain of integration. As an input, we choose the onedimensional x dependent stationary solution with ␭⫽0.5. It
is to be noted that the photon number and the Hamiltonian
are conserved to order 10⫺5 during the calculation. Since theinitial value of the integrated angular momentum is zero,
it is found to always remain zero 共within an error oforder 10⫺15).
Strictly speaking, the eigenfunction in the preceding section 共see Fig. 2兲 contains cores of angular momenta, i.e.,crossings of the real and imaginary zeros of the field due tothe periodic form of the perturbation in the y direction. Thus,
the growth of small amplitude perturbation itself does notexactly mean the ‘‘creation’’ of angular momenta. In order tocheck the real creation of angular momenta from an exact
zero everywhere, we have first carried out the calculation
with an initial condition of the form冋

A 共 x,y,0兲 ⫽ ␣ g 共 x 兲 ⫹Ã 1

册dg
exp共 ⫺ ␤ y 2 兲 ⫹i 冑1⫺ ␣ 2 g 共 x 兲 ,dx
共34兲with ␣ ⫽0.9, ␤ ⫽0.0875, and Ã 1 ⫽0.1. This initial condition
is so arranged that there is no crossing of the real and imaginary zeros. It is done by imposing a real-valued perturbation
on a complex-valued stationary solution. The zero line of theimaginary part exactly coincides with the y axis, while thatof the real part deviates from the y axis by Ã 1 . As the systemevolves, we observe the creation of crossings and the appearance of vortex solitons. As the initial conditions are changed
共always starting form no crossings兲, the main qualitative

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FIG. 4. Crossing of zeros attime t⫽40, 44, 48, and 72.

result—the emergence of angular momentum carrying vortex
soliton chains—is found to be fairly robust.
Next we investigate the evolution of the system undertransverse perturbations of the form

A 共 x,y,0兲 ⫽g 共 x 兲 ⫹Ã

冉 冊

dg
n␲
cos
y ,
dx
L

共35兲

with Ã⫽0.1 and n⫽6. The zero lines of the real and imaginary parts at t⫽2 are shown in Fig. 3. As we noted in Sec. II,
the crossing points of two zero lines correspond to the vortex
centers. We initially have 12 crossings, suggesting 12 vortices. These 12 vortices, however, do not have well-formed
solitonic structures since they are too close and overlapping.
As the system evolves, the vortices move and we seeannihilations of pairs of opposite polarity. The first annihilation event was observed in the interval 28ⱗtⱗ34; two of thevortices destroy one another near the center, and the othertwo seem to disappear near the edge; the annihilation of two
pairs removes four from the original 12 at t⫽2.
The second annihilation event occurs during 40ⱗtⱗ48.
As is depicted in Fig. 4, the zero lines of the real and imaginary parts tend to separate around the center y⬃0. Here two
vortices approach the origin around 40ⱗtⱗ44, and thenthey are annihilated. After the annihilation, two inner vorti-

ces approach the origin. Finally the remaining six vortices
tend to spread and align with equal intervortex spacing 共seet⫽72 in Fig. 4兲.
The amplitude 兩 A 兩 corresponding to this sequence is illustrated in Fig. 5. At t⫽48, the vortices do not quite look likevortex solitons, even though the number of vortices has been
reduced to six. Since the distance between adjacent vortices
is still small, they overlap and are not quite independent. The
central hump around x⬃y⬃0 is the remnant of the second
annihilation event. With time, the peaks tend to expand to thecentral region and are sufficiently apart to look and behave
like vortex solitons. By t⬃72, we observe the formation ofsix solitonic structures as predicted in Ref. 关10兴 共see t⫽72 in
Fig. 5兲. Notice that the vortices disappear only when annihilated by another with opposite polarity.
After tⲏ80, a propagating wave is clearly seen in theregion y⬍⫺10 and y⬎10 where the field was originally flat
共see Fig. 6兲. This propagating field is the trace of the Cherenkov radiation which comes from the nonintegrability ofthe system 关20兴. The radiation will propagate away from vortices. In the finite calculation domain, the radiation will bereflected back at the boundary.
When we extend the time evolution further, we observe
the third annihilation event around t⬃160. However, thisannihilation may be an artifact of the finite size of the domain. When we carry out the simulation in a domain larger

FIG. 5. Amplitude 兩 A 兩 at timet⫽48 and 72.

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FIG. 6. Radiation appears after t⬎80.

than (L⫽28) but with the same initial conditions, the third
annihilation event takes place at a later time, while the timesof the first and the second annihilation events remain unchanged. The third annihilation event is likely to be driven
by the reflection of the Cherenkov radiation at the boundary.
Equivalently, it may be concluded that six vortex solitons arestable in our domain and may stay forever. It is noted that the
distance among vortices at the final stage approximately coincides with the inverse of the wave number with the maximum growth rate k m .

standard nonlinear Schrödinger equation, are unstable totransverse perturbations. By carrying out a linear analysis,
we have found that there exists an instability window oftransverse wave numbers for the system. By a numericalsimulation of the fully nonlinear equation, we have shownthat the transverse instability will yield, after a few annihilation events, a well-separated chain of vortex solitons withalternating, singly charged polarity or topological charge(m⫽⫾1). These singly charged vortex solitons are topologically stable and do not disappear unless they collide with
their complements and annihilate. The number of the createdvortex solitons seem to be determined by the inverse of the
wave number k m with the maximum linear growth rate. For
the box size L⫽28, the six vortex soliton state is found to berobust.
We have suggested a simple and plausible mechanism of
angular momentum generation in the MeV epoch of the Universe. Electromagnetism, operating through the versatile
substrate of the electron-positron plasma, seems to readily
generate highly interesting, long-lived objects, which are capable of carrying large amounts of mass, energy and angularmomentum. Since an initial localization of mass, energy, andangular momentum is precisely the seed that gravity needs
for eventual structure formation, electromagnetism may haveprovided a key element in the construction of the large-scale
map of the observable Universe.

IV. SUMMARYACKNOWLEDGMENTS

We have demonstrated the dynamics of angular momenta
creation in a highly relativistic electron-positron plasma subject to the passage of a strong pulse of electromagnetic fields.
The system is governed by a generalized nonlinear Schrödinger equation with a defocusing inverse-square-root-type
nonlinearity. It turns out that the one-dimensional dark soliton stripe solutions of this equation, just like those of theThis work was supported in part by a Grant-in-Aid fromthe Japanese Ministry of Education, Culture, Sports, Scienceand Technology, Grant No. 12780353. The work of V.I.B.
was partially supported by the ISTC Grant No. G-663, andS.M.M. was supported by the U.S. Department of Energy
Contract No. DE-FG03-96ER-54346.

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